MATERIALS AND METHODS

Transcription

1 Amin e al., The Journal of Animal & Plan Sciences, 24(5): 204, Page: J Anim. Plan Sci. 24(5):204 ISSN: TIME SERIES MODELING FOR FORECASTING WHEAT PRODUCTION OF PAKISTAN M. Amin, M. Amanullah and A. Akbar Deparmen of Saisics Bahauddin Zakariya Universiy, Mulan, Pakisan. Corresponding Auhor E mail: [email protected] ABSTRACT Whea is he main agriculure crop of Pakisan. For counry planning, forecasing is he main ool for predicing he producion of whea o deermine he siuaion wha would be he value of producion coming year. In his research, we developed ime series models and bes model is idenified for he objecive o forecas he whea producion of Pakisan. In his research large ime periods i.e daa was used. Various ime series models are fied on his daa using wo sofware s JMP and Sagraphics. We have found ha he bes model is ARIMA (, 2, 2). On he basis of his seleced model, we have found ha whea producion of Pakisan would become housand ons in 2020 and would become double in 2060 as compared in 200. Key words: ARIMA; Time Series models; Whea Producion Forecasing. INTRODUCTION Agriculure is he backbone of Pakisan s economy and i conribues o he economic and social well being of he naion hrough is influence on he gross domesic produc (GDP), employmen and foreign exchange earnings. In food grain crops, whea and rice are he mos imporan in he agriculure secor, in ha rice conribues 5.4% of value added in agriculure and.3% o GDP and whea accouns for 3.8% in value added in agriculure and 3.4% in GDP (Go vernmen of Pakisan, 2004). Agriculure plays an imporan role in he beermen of he large proporion of he rural populaion in paricular and overall economy in general. Agriculural developmen is desired in almos every par of he world oday. The race beween increasing populaion and food supply is a real grim. Whea is he main saple food for he people of Pakisan. The unprecedened drough and waer shorage condiions have severely affeced he whea crop during he las years. Whea producion forecasing is mainly depends on he culivaed area. Therefore i is necessary o develop he model for, o deermine he esimaed whea producion on he basis of culivaed area in he long run. Many sudies have been conduced o forecas and deermine consrains in he producion of major crops such as whea, coon and rice in Pakisan. Despie hese consrains, here are indeed good prospecs for coninued growh in he area and yield of whea and oher crops in Pakisan ( Hamid e al., 987; Muhammad, 989). Qureshi e al. (992) analyzed he relaive conribuion of area and yield o oal producion of whea and maize in Pakisan and concluded ha here was more han 00% increase in oal whea producion ha can be aribued o yield enhancemen. Muhammad e al. (992) conduced an empirical sudy of modeling and forecasing ime series daa of rice producion in Pakisan. ARIMA model has been frequenly employed o forecas he fuure requiremens in erms of inernal consumpion and expor o adop appropriae measures (Muhammad e al., 992; Shabur and Haque, 993; Sohail e al., 994). Facually, oher crops in general and whea in paricular provide linkages hrough which i can simulae economic growh in oher secors. Whea culivaion has been suffering from various problems, such as radiional mehods of farming, low yields, shorage of key inpus and shorage of irrigaion waer. Pakisan has experienced ups and downs in whea producion. Prices of whea and flour booss up during low producion seasons and falls drasically when here is a surplus whea producion, however, surplus whea producion occurred for few years and during such periods farming communiy suffered heavy losses due o inadequae markeing faciliies in he counry. On he oher hand, farmers do no know fuure prospec of whea producion and prices while deciding o culivae his and oher crops. There is a dire need o forecas area, yield and producion of whea in Pakisan. Therefore, he objecive of his paper is o deermine fuure prospecs of whea in he counry using pas rends. Karim e al. (2005) applied regression modeling o forecas whea producion of Bangladesh disrics. They used seven model selecion crieria s and found ha differen models were idenified for differen disrics for whea producion forecass. They have found ha whea producion in Bangladesh disrics i.e. Dinajpur, Rajshahi, and Rangpur would be.54, 0.35, 0.3, and 0.58 million ons, respecively, in 2009/0. Iqbal e al. (2000) used 444

2 Amin e al., J. Anim. Plan Sci. 24(5):204 ARIMA model for forecasing whea area and producion in Pakisan. They used ARIMA (,,) model for whea area forecasing and ARIMA(2,,2) model for whea producion forecasing. They have found ha for forecass of whea area was abou housand hecares. A whea area forecas for he year 2022 was housands hecares. Forecass of whea producion showed an increasing rend. For , a forecas of whea producion was abou housands ons while whea producion forecas for he year 2022 came o be abou housand ons. Boken (2000), in Canadian Prairies were applied differen ime series models on whea yield o forecas spring whea yield. He used MSE as deerminisic crieria o selec he bes model and found ha quadraic model is bes for whea yield forecasing. While on he basis of sochasic crieria, he found ha simple average model is bes. Saeed e al. (2000) has also applied ime series model o forecass whea producion of Pakisan. They used he whea producion daa series from o bu hey have no menioned he source of daa. They suggesed ARIMA (2,2,) model for whea forecas of Pakisan, on he basis of his model hey furher forecas whea producion for 5 years i.e o For , hey prediced ha foresed whea producion is million ons. Schmiz and Was (970) were applied ime series modeling o predic whea yield of four counries Canada, Unied Saes, Ausralia and Argenina for he period 950 o 966. They compare parameric ime series modeling and smoohing and conclude ha rend models are bes for yield forecasing. Sabir and Tahir (202) forecas he whea producion, area, and populaion for he year 20-2 by using exponenial smoohing. They have found ha he need of whea is 2.70 million ons for he populaion of million for he year All hey sudied and forecas he whea producion on he basis of small ime periods daa and no given any discussion abou he assumpions of seleced ime series model. The objecive of he curren sudy is o forecas he whea producion of Pakisan on he basis of large daa and ime series model wih cerain model assumpions are hold for beer planning o improve he producion o fulfill he demand of Pakisan naion. MATERIALS AND METHODS Respecive ime series daa for his sudy were colleced from Governmen Publicaions such as Agriculural Saisics of Pakisan and Pakisan Economic Survey. For forecasing purposes, various models are available and we are seeking for he bes one. Box and Jenkins (976) linear ime series model are applied in our research for forecasing whea producion o mee he challenges i.e. shorage of whea in advanced. Auoregressive Inegraed Moving Average (ARIMA) is he mos general class of models for forecasing a ime series. Differen series appearing in he forecasing equaions are called Auoregressive process. Appearance of lags of he forecas errors in he model is called moving average process. The ARIMA model is denoed by ARIMA (p,d,q), where p sands for he order of he auo regressive process, d is he order of differencing and q is he order of he moving average process. Some of our sudy ineres ARIMA models wih reference are given in able ; Table. Forms of ime Series models for whea producion of Pakisan Sr. No Models Name Model Equaion Reference Random Walk Wih Drif 0 y y e Casella, e al., Linear Trend 4 Simple Exponenial Smoohing 5 ARIMA (0,, ) = IAM (, ) y wih consan y 0 e e 6 ARIMA (0,, ) = IAM (, ) y a b e Casella, e al., 2008 yˆ y y Casella, e al., 2008 Casella, e al., 2008 y y e e Casella, e al., 2008 y y e e e Casella, e al., ARIMA (0,, 2) wih consan y y y e e Casella, e al., ARIMA (,, ) wih consan 2 0 y y y e e Casella, e al., ARIMA (,, ) 2 y 2y y e e e Casella, e al., ARIMA (0, 2, 2) wih consan ARIMA (0, 2, 2) y 2y y2 e e 2e2 Casella, e al.,

3 Amin e al., J. Anim. Plan Sci. 24(5):204 For more forms of ime series models and parameer esimaion in deail see (Casella e al., 2008; Tsay, 2005; Chafield, 995). In ARIMA modeling, he order of AR(p) is idenified by parial auocorrelaion funcion (PACF) while he order of MA(q) is idenified by auocorrelaion funcion (ACF) (Tsay, 2002). The order of ARIMA (p, d, q) is also idenified by model selecion crieria s i.e. Schwarz Bayesian informaion crieria (SBIC) and Akaike s Informaion Crieria (AIC) (Casella, e al., 2008). These crieria s are furher explained in model specificaion secion. Model Specificaion: One of he imporan issues in ime series forecasing is o specify model. Time series model is specified on he basis of some informaion crieria s which includes AIC, BIC likelihood ec. Akaike s (973) inroduced AIC crieria for model specificaion. AIC is mahemaically defined as; AIC 2log max imum likelihood 2k Where k = p+q+ (if model includes inercep) oherwise k = p+q. model specified well if is AIC value is minimum as oher fied models (Tsay, 2005). Oher model specificaion crierion is SBIC and is compued as; SBIC 2 log max imum likelihood 2k log n. Model which has minimum SBIC value specified well as oher fied models (Tsay, 2005) Time Series Model Diagnosics: The ime series model assumpion includes independence, normaliy, auocorrelaion ec of residuals of he bes fied models. Auocorelaion is esed by Runs Tes (Gujarai, 2004) and Box- Pierce es developed by Box and Pierce (970) and ACF and PACF are also used o deec he auocorrelaion in he daa (Elivli e al., 2009). Residual normaliy is esed hrough normal probabiliy plo, and residual inegraed periodogram which displays Kolmogorov-Smirnov 95 % and 99 % bounds. If he residuals are random hen periodogram fall wihin hese bounds, which is also an indicaion of whie noise of residuals (Casella e al., 2008). Forecasing Accuracy Measuring Techniques: Afer model selecion, a nex imporan sep is o measure he accuracy o verify he reliabiliy of forecased value based seleced model. Various ools are available in lieraure which includes Roo mean square error (RMSE), mean absolue error (MAE), mean absolue percenage error (MAPE), mean error (ME) and mean percenage error (MPE). Furher compuaion and lieraure of hese accuracy measuring ools are given in able 2; Table 2. Forecass accuracy measuring ools Accuracy measuring ool MAE ME MSE MPE MAPE Where value for ime. Formulaion MAE ME MSE MPE MPE n n n n e e 2 n e n n PE n n PE n Reference Makridakis e Makridakis e Makridakis e Makridakis e Makridakis e Y F PE 00 Y and F is he forecased RESULTS AND DISCUSSION In his research, ime series models were fied on whea producion daa of Pakisan. The objecive of fiing muliple ime series models on his daa is o obained reliable forecass on he basis of saisical measures. Whea producion Forecasing In able 4, differen ime series models fied and resuls are presened wih model selecion and validiy crieria s. On he basis of AIC we have found ha Model M i.e. ARIMA(, 2, 2) has lowes AIC and we use his model o forecas whea producion of Pakisan on he basis of hisorical daa i.e In his able we also summarize he resuls of five ess run on he residuals o deermine wheher each model is adequae for he daa. Noe ha he currenly seleced model, model M, passes 4 ess. Since no ess are saisically significan a he 95% or higher confidence level, he curren model is probably adequae for he daa. Similar resuls are also obained by using JMP Sofware as on he basis of AIC and SBC ranks he bes seleced model is ARIMA (, 2, 2) and he resuls are shown in able 3. The ARIMA(,2,2) model coefficien summary is given in able

4 Amin e al., J. Anim. Plan Sci. 24(5):204 Table 3. Model Comparison Using JMP8 of Whea Producion Forecasing of Pakisan Model Comparison Using Sagraphics 5 Models (A) Random walk wih drif = (B) Consan mean = (C) Linear rend = (H) Simple exponenial smoohing wih alpha = (I) Brown's linear exp. smoohing wih alpha = 0.97 (J) Hol's linear exp. smoohing wih alpha = 0.2 and bea = (M) ARIMA(,2,2) (N) ARIMA(0,2,2) (O) ARIMA(,,2) (P) ARIMA(0,2,2) wih consan (Q) ARIMA(,0,) wih consan Table 4. Model Selecion and validiy model esing crieria s of Whea producion Forecasing based on Model ermse MAE MAPE ME MPE AIC RMSE RUNS RUNM AUTO MEAN VAR (A) E OK OK OK OK ** (B) E * *** *** *** *** (C) E * *** *** OK ** (H) OK OK OK ** *** (I) * OK OK OK ** (J) OK OK OK OK *** (M) OK OK OK OK *** (N) OK OK OK OK *** (O) OK OK OK OK *** (P) OK OK OK OK ** (Q) OK OK OK OK ** Table 5. ARIMA (,2,2) Model Coefficien Summary Parameer Esimae Snd. Error P-value AR() MA() MA(2) On he basis of Table 5, model coefficiens he esimaed whea forecased model is; yˆ yˆ ˆ.85085ˆ e e2 ŷ Where is he forecased whea producion for ime years. yˆ is he forecased whea producion of one previous year ˆ e is he previous one year residual as indicaed in appendix able ˆ 2 e able is he previous wo year residual as indicaed in appendix Tesing Seleced Model Assumpions (Normaliy, Auocorrelaion and Heeroscedasiciy): We ge he reliable whea producion fuure value if he seleced model is good. Seleced model is good one, if i fulfills 447

5 Amin e al., J. Anim. Plan Sci. 24(5):204 he assumpions i.e. Normaliy, Auocorrelaion and Heeroscedasiciy of he seleced model residuals. Table 6. Tess for Auocorrelaion and Independence Tes Tes Saisic p-value Value Runs above and below median Runs up and down Box-Pierce Tes Table 6, indicaed ha he ARIMA (, 2, 2) model residuals are uncorrelaed as well as independen as all hree ess signify. Three ess have been run o deermine wheher or no he residuals form a random sequence of numbers. A sequence of random numbers is ofen called whie noise, since i conains equal conribuions a many frequencies. The firs es couns he number of imes he sequence was above or below he median. The number of such runs equals 49, as compared o an expeced value of 52 if he sequence were random. Since he P-value for his es is greaer han or equal o 0.05, we canno rejec he hypohesis ha he residuals are random a he 95.0% or higher confidence level. The second es couns he number of imes he sequence rose or fell. The number of such runs equals 68, as compared o an expeced value of if he sequence were random. Since he P-value for his es is greaer han or equal o 0.05, we canno rejec he hypohesis ha he series is random a he 95.0% or higher confidence level. The hird es is based on he sum of squares of he firs 24 auocorrelaion coefficiens. Since he P-value for his es is greaer han or equal o 0.05, we canno rejec he hypohesis ha he series is random a he 95.0% or higher confidence level. The normaliy is also esed by normal probabiliy plo as shown in figure () and periodogram as shown in figure (4) boh figures indicaed ha he residuals of ARIMA (, 2, 2) are normally disribued. The heeroscedasiciy is esed by Var es as show in Table 3 indicaed ha ARIMA (, 2, 2) residuals are heeroscedasic. There is no indicaion of auocoorelaion in residuals of seleced model signifies by runs and Box-Pierce es Residual Normal Probabiliy Plo ARIMA(,2,2) 2300 Residual percenage Figure. Residuals Normal Probabiliy Plo of Whea Producion Model for Residual Auocorrelaions for Whea Producion ARIMA(,2,2) Auocorrelaions lag Figure 2. Residuals Auocorrelaion Plo of Whea Producion of ARIMA (, 2, 2) Model 448

6 Amin e al., J. Anim. Plan Sci. 24(5):204 Residual Parial Auocorrelaions for Whea Producion ARIMA(,2,2) Parial Auocorrelaions lag Figure 3. Residuals Parial Auocorrelaion Plo of Whea Producion of ARIMA (, 2, 2) Periodogram for Residuals 0.8 Ordinae frequency Figure 4. Periodogram of Residuals for Whea Producion ARIMA (, 2, 2) Model Table 7. One sep ahead forecass and residuals for whea producion daa ( ) year Daa Forecas Residual Table 8. Whea producion forecass (in housand ons) wih inerval of 0 years Year Forecas One sep ahead forecass and residual of whea producion on he basis of 902 o 2005 whea producion daa for he year 2006 o 200 is presened in able 7. Whea producion forecased value on he basis of ARIMA (,2,2) model wih inerval of 0 year from 2020 o 2060 is presened in able 8. From able 6, we have found ha whea producion of Pakisan would become housand ons in 2020, in 2030, in 2040, in 2050 and housand ons in As he forecasing is based on sound saisical formulaion, so i is adequae forecasing provided ha he environmenal condiions remain same. 449

7 Amin e al., J. Anim. Plan Sci. 24(5):204 Conclusion: As he populaion increases over ime gradually, herefore i is necessary o plan o mee he requiremens of naion. For his purpose, forecasing is he key ool o alarm abou he need of naion in advance. Whea is he basic need of any counry all over he world. In his sudy, we developed ime series models o forecass whea producion of Pakisan on he basis of hisorical daa i.e We have developed differen ime series models on whea producion of Pakisan on his daa. Bes model is seleced on he basis of model selecion crieria i.e. AIC and SBIC. Main ineres of developing ime series model as oher sudies is ha he model fied is also saisfied residual assumpions i.e. normaliy, independence and no auocorrelaion. On he basis of hese model selecion crieria, we have found ha bes model for whea producion forecasing of Pakisan is ARIMA (, 2, 2). On he basis of developed ime series model, we have found ha bes ime series model for forecasing whea producion of Pakisan is ARIMA (, 2, 2) because his model has lower AIC and SBIC as compared o oher fied ime series models. On he basis of his model, we have found ha whea producion of Pakisan would become housand ons in 2020 and would become double in 2060 as compared in 200 under he assumpion ha here is no irregular movemen or variaion is occurred. REFERENCES Boken, V.K. (2000). Forecasing spring whea yield using ime series analysis: a case sudy for he Canadian Prairies. Agron. J, 92: Box, G.E.P. and G.M. Jenkin (970). Time Series Analysis, Forecasing and Conrol. (San Francisco: Holden-Day,). Box, G.E.P. and D.A. Pierce (970). Disribuion of residual auocorrelaions in auoregressiveinegraed moving average ime-series models. J. Amer. Sa. Asso. 65: Box, G.E.P., G.M. Jenkins and G.C. Reinsel (994). Time series Analysis, Forecasing and Conrol. 3rd Ed. Englewood Cliffs, N J: Prenice-Hall. Chafield, C. (995). The Analysis of Time Series an Inroducion. 5 h Ed. Chapman & Halljcrc Boca Raon London New York Washingon, D.C. Elivli, S., N. Uzgoren and M. Savas (2009). Conrol chars for auocorrelaed colemanie daa. J.Sci. Indus Res. 68: -7. Governmen of Pakisan. (2004). Economic Survey Finance Division. Economic Adviser s Wing, Islamabad. Hamid, N.P., V. Thomas, Albero and G. Suzanne (987). The whea economy of Pakisan seing and prospecs. Inernaional Food Policy Research Insiue, Minisry of Food and Agriculure, Governmen of Pakisan, Islamabad, Pakisan. Iqbal, N., K. Bakhsh, K. Maqbool, and A.S. Ahmad (2000). Use of he ARIMA model for forecasing whea area and producion in Pakisan. In. J. Agri. Biol. 2: Karim, R., A. Awal and M. Akher (2005). Forecasing of whea producion in Bangladesh. J.Agri. Soc. Sci. : Makridakis, S., S.C. Wheelwrigh and R.J. Hyndman (2003). Forecasing: mehods and applicaions. 3 rd Ed. John Wiley and Sons. Muhammad, K. (989). Descripion of he hisorical background of whea improvemen in baluchisan, Pakisan. Agriculure Research Insiue (Sariab, Quea, Baluchisan, Pakisan). Muhammad, F., M. Siddique, M. Bashir and S. Ahmad (992). Forecasing rice producion in Pakisan using ARIMA Models. J.Anim.Plan.Sci. 2: Sabir, H.M. and S.H. Tahir, ( 202). Supply and demand projecion of whea in Punjab for he year Inerdis. J. Conemp. Res. Bus. 3: Saeed, N., A. Saeed, M. Zakria and T.M. Bajwa (2003). Forecasing of whea producion in Pakisan using arima models. In. J. Agri. Biol. 2: Tsay, R.S. (2002). Analysis of Financial Time Series: Financial Economerics. John Wiley & Sons, Inc. Qureshi, K., A.B. Akhar, M. Aslam, A. Ullah anda. Hussain ( 992). An Analysis of he relaive conribuion of area and yield o Toal producion of whea and maize in Pakisan. J. Agri. Sci. 29: Shabur, S.A. and M.E. Haque (993). An analysis of rice price in Mymensing own marke paern and forecasing. Bang. J. Agri. Econo. 6: Sohail, A., A. Sarwar and M. Kamran (994). Forecasing oal food grains in Pakisan. J. Engi. Appl. Sci. 3:

8 Amin e al., J. Anim. Plan Sci. 24(5):204 Appendix ARIMA (, 2, 2) Period() Whea Forecased value of Residual Producion whea producion ê y ŷ